Real-Time Simulation of Heat Transfer in Continuous Casting SEPPO LOUHENKILPI, ERKKI LAITINEN, and RISTO NIEMINEN A real-time heat-transfer model for continuous slab casting is presented. The model calculates the strand temperatures and the solid shell thickness profile along the machine as a function of the actual casting variables, strand geometry, and steel grade. The special requirements concerning the real-time use of the model and, in general, the accuracy of the model are also studied and discussed. The model has been tested by carrying out industrial trials. Some examples of the differences between the calculated and measured surface temperatures are presented. A special procedure to determine the boundary conditions for the secondary cooling zones from temperature measurements is also described.

I.

INTRODUCTION

THE careful control of the strand cooling and the shell growth along the machine is of central importance in continuous casting operation. These have a considerable influence on the formation of cracks and other defects which can be formed in the cast material. To ensure defectfree products, the strand is to be cooled down according to a pattern which depends on steel grade, product dimension, casting speed, and machine design. On the other hand, the control of the liquid pool length is a key element in optimizing the casting speed with respect to good productivity. So, the heat transfer plays a very important role in continuous casting, especially when casting cracksensitive steel grades. To study the thermal state of a continuously cast strand, two methods can be used: empirical correlation of numerous experimental results and mathematical simulation models supported by experimental results. It is difficult and inaccurate to measure, at least inside the spray chamber, the strand temperatures or the shell thickness during casting. Moreover, the empirical models cannot be used to extrapolate the results outside the experimental range and generally they cannot be used for simulation of transient casting conditions. Mathematical models, on the other hand, once verified are easy to use and comprehensive in simulating the thermal state of the strand. In recent years, many mathematical heat-transfer models for continuous casting have been developed, t~-2~l However, most of the models can be used only for simulation of steady state casting operations in off-line. They give the strand temperature field as a function of casting parameters, such as the casting speed, superheat, mold heat removal, spray water flow rates, steel grade, and strand geometry. The numerical approximation of the model is usually done by the finite difference or finite element method. Heat-transfer models are being increasingly used to improve the existing cooling systems, to improve casting practices, and for process control. SEPPO LOUHENKILPI, Senior Teaching Assistant, is with the Laboratory of Metallurgy, Helsinki University of Technology, 02150 Espoo. Finland. ERKKI LAITINEN, Associate Professor, and RISTO NIEMINEN, Researcher, are with the Department of Mathematics, Jyv~iskylfi University, 40100 Jyv~iskyl~i, Finland. Manuscript submitted October 13, 1992. METALLURGICAL TRANSACTIONS B

Although the steady state models offer important knowledge for the operational limits of the continuous casting machine, they are not valid for simulation of transient casting conditions which occur rather frequently. For better control of heat transfer over the whole continuous casting cycle, more attention has recently focused on developing real-time simulation models which are valid under transient casting conditionsJ ~6-211 In the real-time simulation, many practical requirements will be set for the simulation model. The computing time must, for instance, be short enough and the special process conditions, such as the start and the end of casting, must be included in the model. In the present article, a real-time heat-transfer model for continuous casting is presented. The model calculates the strand temperature field in a longitudinal cross section through the middle of the strand. The longitudinal heat conduction is taken into account; therefore, the model can be applied also for the simulation of continuous casting of copper, aluminum, and other metals where this factor is significant. The numerical approximation of the heat-transfer model is based on the finite element method. For the correct simulation of heat transfer in continuous casting, the determination of the boundary conditions describing the heat-transfer phenomena through the strand surface is of crucial importance. The boundary conditions are usually expressed as heat fluxes or heattransfer coefficients. A procedure to determine the boundary conditions for the secondary cooling zones from the temperature measurements is presented in this article. For the validation of the heat-transfer model, a lot of strand surface temperature measurements were carried out on a slab casting machine and the measured temperatures were compared with the calculated ones. The special requirements concerning the real-time use of the model and, in general, the accuracy of the model are also studied and discussed. II.

THE REAL-TIME SIMULATION MODEL

The model simulates the temperature field in a longitudinal cross section through the middle of the strand. The calculation domain is schematically shown in Figure 1. The model is two- dimensional (2-D), the dimensions being the casting direction and the direction VOLUME 24B, AUGUST 1993--685

perpendicular to the wide face of the strand. The heat flow along the width direction of the strand is neglected. In the case of rectangular sections, such as slabs and thin sections, this simplification is generally accepted, but for blooms or billets, the model cannot be directly applied. The other typical assumptions made in the model are that (a) the solidus and liquidus temperatures as well as other phase transformation temperatures are constant, (b) the solidification takes place by directional dendritic growth, (c) the material behavior is isotropic, and (d) the cooling is symmetrical on the top and bottom sides of the slab. These assumptions are generally made also in steady state models. Heat transfer within this 2-D domain can be mathematically defined by

OH(T) Ot

+ v(t)

OH(T) Oy

- V. (k(T)VT)

[1]

where H is the enthalpy, t is time, k is thermal conductivity, T is temperature, and v is the actual casting speed. Enthalpy H is defined as the sum of sensible and latent heats:

H(T) = Ah(T) + AH(T)

[2]

The effect of the form of fs on the calculated results is described in Section VI. If there are other phase transformations in the solid state, such as the A-ferrite to austenite and the austenite to a-ferrite, they can be taken into account in a similar way. Applying Kirchhoff's transformation,

I

c(O) dO

[3]

and (" T

AI-I(T) = pL Jo

Ofso0dO = pL (1 - f s ( T ) )

[4]

where p is density, c is specific heat, L is latent heat, and f, = f~(T) is solid fraction. The solid fraction describes the manner in which the latent heat is released between the solidus and liquidus temperatures. This manner is very sensitive to the chemical composition of the metal being cast and is not always known accurately.

kdO

[5]

Eq. [1] becomes:

OH ( 9 2 K ~ - Ot Ox2

--

OaK -Oy2

-}- - -

8H Oy

[6]

V--

The last term in Eq. [6] represents the convective heat transfer due to the movement of the strand. The convective heat transfer in the liquid due to the fluid flow is difficult to calculate using differential equations. The technique most often used to account for the convective heat transfer is effective thermal conductivity. t',2z.23j This technique is also used in the present case. The effective thermal conductivity is approximated by the following linear relationship:

where the sensible heat, Ah(T), and the latent heat, AH(T), are defined as

Ah(T) = p

fo

K(T) =

keff(Z) -~ k ( T ) f s

+ Ak(T)

(1 - f ~ )

[7]

where k is thermal conductivity and A is a parameter. If the parameter A is 1, there is no increased heat transfer in the mushy or liquid regions due to the fluid flow. Figures 2 and 3 show calculated shell thicknesses and slab surface temperatures with A = 1 and A = 5. The parameter A has more influence on the results within the liquid pool than in the solid shell. Normally, a value of 5 is used in our calculations. Mizikar 122Jreported that the empirical keff was roughly 7 times greater than the liquid thermal conductivity, and this approach is generally used in solidification simulations. However, Mizikar used the effective thermal conductivity only over the region where the temperatures were over the liquidus temperature. In the present model, the effective thermal conductivity is used also over the mushy region, i.e.,

x-a~

MOLD

Liquidus .

t tt

~

,

~

=

-"

= A=I

5

6

7

0 0

1

2

3

4

8

9

Distance from meniscus, m Fig. 1 - - T h e calculation domain schematically presented. 686--VOLUME 24B, AUGUST 1993

Fig. 2 - - S o l i d u s and liquidus isotherms calculated with A = 1 and A = 5 (To = 1500 ~ TL = 1454 ~ and Ts = 1399 ~ METALLURGICAL TRANSACTIONS B

where the temperature is between the liquidus and solidus temperatures (Eq. [7]). That is the reason why a little smaller value of constant A is used in our calculations.

III.

BOUNDARY

AND INITIAL CONDITIONS

In order to solve Eq. [6], the following boundary and initial conditions are used. (1) During steady casting, the liquid surface at the top of the mold is the upper boundary of the calculation domain. Here, the temperature is assumed to be equal to the incoming liquid temperature To = To(t). Hence, [8]

T(x, O) = To

During the end of the cast, this boundary descents with the strand according to the actual casting speed and radiative heat flux is used as a boundary condition. Thus, 8T - k - - = ~e(T 4 - T~xt) during the end of the cast On

[9]

(2) The convective energy transport due to the strand movement, Q = Q(x; t), is used as the boundary condition at the lower boundary of the domain: Q =- vH [10] (3) At the strand surface along the mold length, the heat flux through the mold wall, Q = Q(y; t), is used as a boundary condition. The average heat flux, Q .... extracted from the mold is calculated from the heat balance of the mold cooling water:

AT Q,ve = p w c w W - Am

OT 4 - k - - = h(T -- Tw) + O'E(T4 -- T~xt) On

OT -k-- = Q On

[121

The last term represents the heat flux by radiation. Before this boundary condition can be applied in industrial computations, the relationship between the effective heattransfer coefficient and the cooling parameters, e.g., spray water flow rates, strand surface temperatures, etc., must be determined for each cooling zone. Approximate values can be found in the literature, 17-]~ but for a specific caster and steel grade, these can be determined more precisely on the basis of surface temperature measurements, for instance, using pyrometers or thermoelements. This can be done by using, for instance, a method which is presented in Section VII.

1200

1400

O

41

lo0o

o r

heat flux f i t t e d

Mold

12oo

4--*

O O

[13]

[I 1]

Here, AT is the temperature increase of the mold cooling water, Am the area of the mold face, and W the cooling water flow rate. The heat flux distribution along the mold length, Q, is then calculated by fitting the average heat flux to experimentally obtained heat flux profiles. 16,24,251 Hence, the heat flux through the mold wall can be expressed as

~o

The total amount of heat flux can be easily measured, but the heat flux profile varies with casting parameters, e.g., with casting speed and the steel grade to be cast. However, this profile is not always very well known. To investigate how sensitive the simulation model is to the changes of the mold heat flux profile, test calculations were carried out. Strand surface temperatures calculated with two different mold heat flux profiles are presented in Figure 4. The total amount of the mold heat flux, Q .... was the same. In the fitted case (Figure 4), the air gap formation was approximated by decreasing the heat flux at the bottom of the mold. In the unfitted case, the heat flux was distributed uniformly along the mold. As can be seen (Figure 4), the different approximation of the heat flux distribution profile has a minor effect on the temperatures in the solid shell below the mold. Inside the mold, however, the influence is considerable. So, the correct assignment of the mold heat flux distribution profile is important only when studying temperatures in the mold region or immediately below it. (4) At the strand surface below the mold, the heat flux from the strand surface to the environment takes place by a combination of the three heat-transfer mechanisms: conduction, convection, and radiation. It is very difficult to determine the shares of each mechanism on the total heat transfer. In the present model, an effective heattransfer coefficient, h, is used. This takes into account the total heat transfer except the radiation. Hence, the boundary condition is

~1ooo 0

900 8O0 0

t

I

2

4

6

i

I

8

10

12

Distance from meniscus, m Fig. 3 - - Strand surface temperature profile calculated with A = 1 and A = 5 (To = 1500~ TL = 1454~ andTs--- 1399~ METALLURGICAL TRANSACTIONS B

o

800

U)

6O0

. _

Mold heat flux unfitted

........

~ .....

(uniform heat flux in the mold) i

i

i

i

2

4

6

10

12

D i s t a n c e from m e n i s c u s , m Fig. 4 - - S t r a n d surface temperatures for uniform (=unfitted) and for nonuniform (=fitted) heat flux from the mold. VOLUME 24B, AUGUST 1993--687

If local variations in the temperature distribution within a cooling zone are of interest, the effective heat-transfer coefficient must be determined locally in more detail. (5) Because the cooling is assumed to be equal on the top and bottom surfaces, there is no heat flux through the centerline of the slab. Hence, the boundary condition through the centerline is OT

-k--

= 0

The initial condition at the time t = 0 is defined as IT0, = t.Tst~ '

y=0 y > 0

[151

where Tsta~ is a given temperature without any physical importance. Equations [6] through [15] are solved numerically using the finite element method in space and finite difference method in time. In the finite element method, the partial differential equations to be solved are first transformed to algebraic matrix equations. They are then solved using iterative matrix solution methods. The solution of the finite element matrix equation gives the unknown temperatures at nodal points. IV.

Pe -

NUMERICAL SOLUTION EQUATIONS

OF THE MODEL

where Pe p v d k

= = = = =

OH

[g] at where [M] [C] [A] [B] F T K

k

> 2

[19]

discrete Peclet number; density; fluid velocity; element size; and thermal conductivity.

Equation [19] indicates that the size of finite elements must be small if the casting speed is large. In the case of continuous casting, this condition leads, in general, to a very refined element model and, hence, to high computing times. It is very difficult to fulfill this condition in real-time simulation, because the calculation cycle must be short. One method to overcome this is the upwind method. The essential point of the upwind technique is the additional term of artificial conductivity in the convection term. The form of this term depends on the velocity, specific heat, and the element size of the finite element model. In the case of continuous casting, the artificial conductivity may be added along the casting direction. The additional artificial conductivity kaC is defined by t291

In the present work, standard triangle elements with linear shape functions are used. The following finite element matrix equation results from the standard piecewise linear space discretization of Eqs. [6] through [15]: --

pcIvld

[14]

On

T(x,y)

An implicit finite difference scheme is used, because the explicit method places much stricter requirements on the time step size. However, the normal piecewise linear finite element approximation, as in Eq. [ 18], becomes unstable when t291

+ [C]H + [A]K + [ B ] T = F

[16]

= = = = = = =

mass matrix; convection matrix; stiffness matrix; boundary condition matrix; load vector; (T1, /'2, . . . , T,) = temperature vector; (K(Tt), K ( T O . . . . . K(T~)) = Kirchoff's transform vector; and H = ( H ( T , ) , H(T2) . . . . . H(Tn)) = enthalpy vector.

For time discretization, it is denoted OH

H i+l - H i

at

At

[17] where the superscripts i and i + 1 refer to time events t = t and t = ti+~, respectively, and At is the time step between t ~ and {+1. By applying an implicit finite difference scheme, Eq. [16] can be rewritten as

1

k ac = a -~ ( p c v d ) 3

[20]

where a is the parameter (0 -< a --< 1). The representative element size d is, in the case of triangular elements, the length of the longest side of the element. If the parameter a is 0, then no artificial conductivity is applied. Now Eq. [18] can be modified by adding the artificial conductivity term into the convection term:

[M]

--H At

i+l + ([C] + [AaC])Hi+l + [A]K i+1

+ [B] T'+~ = Fi+~ + [M] Hi At

[21]

where [Aac] is the artificial conduction matrix. More details on the upwind method can be found in References 29 through 31. Equation [21] is now the discretized model equation. It is solved for temperature using iterative methods. Details of the solution methods are given in References 32 through 34.

V.

ON THE SPEED OF THE

NUMERICAL SOLUTION ALGORITHM

]M]

_ _ H i+l + [C]H i+l + [A]K i+1 + [B]T i+l At = Fi+l + [M] Hi

At 688--VOLUME 24B, AUGUST 1993

]181

The accuracy of the numerical solution depends mainly on the FE-discretization of the domain. The solutions are comparatively independent of the choice of the discretization as long as the FE-discretization parameters are METALLURGICAL TRANSACTIONS B

sufficiently small. The FE-element mesh used also influences the calculation time which, in real-time simulation, must be short. The Fe-mesh size is a compromise between the accuracy of the solution and calculation time. For testing purposes, the simulation model was solved on a microcomputer and two transputer processors in parallel by using the pipeline technique, p51 The simulation program runs on the two transputer processors on a motherboard plugged into the microcomputer, while the user interface of the program runs simultaneously on the micro's own processor under the Windows environment. In order to study the effects of the time step size and the convergence criterion on the computing time, test trials were carried out. In the test cases, the length of the calculation domain was chosen to be 20.0 m the thickness to be 85 mm. The FE-mesh size was, on average, 10 m m in the thickness direction and 50 to 200 m m in the longitudinal direction, and the total number of nodes was 1540. In Figure 5, a so-called relative computing time is presented as a function of time step. In real-time simulation, the relative computing time (absolute computing time/ length of time step) must be much smaller than 1. For instance, using a 5 second time step, the relative computing time is about 0.4, which means that the absolute computing time is about 2 seconds. When the time step becomes shorter, the absolute computing time becomes shorter also, because the changes in temperature variations between time steps decrease. However, the computing time decreases less than the time step, and thus, the relative computing time increases. In a normal continuous casting machine, a calculation cycle of 5 seconds is believed to be short enough. This can be fulfilled quite easily with the presented installation. The convergence criterion of 0.2~ was used in these trials. Figure 6 shows the relation between the number of iterations and the convergence criterion.

heat, L, and solid fraction, fs(T), necessary for the simulation, literature data w e r e u s e d . [36'37,381 However, it is not always possible to find all the data for a steel grade close enough to the chemical composition to be cast. This is especially the case forf~(T) which can be a strong function of temperature. Test calculations with linear and nonlinear shapes offs were carried out to study the effect of fs on the calculation results. Figure 7 presents some results. The shape does not affect the results in the solid shell significantly but has more influence on the results in the liquid pool. In the nonlinear case, fs was calculated by the computer program developed by Miettinen. p91 The steel grade used was AISI 304L. The nonlinear solid fraction calculated is presented in Figure 8. Normally, a nonlinear f~ is used in our calculations. VII. DETERMINATION OF HEAT-TRANSFER COEFFICIENTS FOR THE SPRAY COOLING ZONES In the case of normal spray water cooling, the heattransfer coefficient depends mainly on the spray water

11 ta~ e-

10

i.................................

9 8 7 11

E

6

z

5 4

0.0

VI.

MATERIAL PROPERTIES

I

I

i

i

I

I

I

0.4

0.6

0.8

1.0

1.2

! .4

1.6

Convergence

For thermophysical properties, such as thermal conductivity, k(T), specific heat, c(T), density, p(T), latent

! .8

c r i t e d a for iterations, ~

Fig. 6 - - A n e x a m p l e of d e p e n d e n c e of n u m b e r of iterations on conv e r g e n c e criterion (time step, 5 s).

1.6 9 E 9'-(3) C :3 Q. E 0

i

0.2

80

"

"

1

/ u~

1.4

J - K

1.2

I=: 6 0

1.0

U[ ca o r-

/

//

/

/

"~

S

SOlidus isotherms

zr

o.8

0

o 0.6

e ._>

"G) ,'-20 0')

0.4

e N 11,2

. . . . .

0.0

,_ _ - , .

I

I

2

I

3

- - a _ _ _,_ . _,_ _ _ L - - - ' . .

l

4

I

5

I

6

I

7

I

8

-'-

=

=

A

0

I0

-"

fs non-linear linear

f S

(solid)

- -

I

9

~

I

t

I

I

2

4

6

8

10

Distance from meniscus, m

L e n g t h of t h e t i m e s t e p , s ~ Fig. 5 - - A n e x a m p l e o f d e p e n d e n c e o f relative c o m p u t i n g time on length of t i m e step ( c o n v e r g e n c e criterion, 0.2 ~ METALLURGICAL TRANSACTIONS B

Fig. 7 - - S o l i d u s and liquidus i s o t h e r m s c a l c u l a t e d w i t h l i n e a r and n o n l i n e a r shapes o f fs. The n o n l i n e a r shape is c a l c u l a t e d for A I S I 3 0 4 L (Fig. 8). VOLUME 24B, AUGUST 1993--689

flow rate and on the surface temperature of the strand. Before the developed heat-transfer model can be applied in industrial computations, the relationship between the heat-transfer coefficient and the spray cooling parameters must be determined separately for each cooling zone. In the present work, empirical formulas for the heattransfer coefficient were derived for a slab casting machine. The caster was of curved type with a straight mold, and the secondary cooling consisted of five zones with normal water spray cooling. The strand temperatures were meastLred with pyrometers under steady and unsteady state casting conditions. Infrared pyrometers were located at 2.7 and 5.8 m below the mold. The temperature was also measured near the straightening unit. The measurements were carried out on both the bottom and top sides of the strand. For other sections of the strand, empirical formulas were estimated from the literature. The following equations were used to describe the relationship between the heat-transfer coefficients, the spray water, and the slab surface temperature: 60

(zones 1 and 2) m~"/--~/fin ' Wi <--60 - (zones 3 and 4) m e min'

(oliW i -~ [3i)%(T) Wi >

hi -~ l

t

aiW~iiYi(T)

t = t ~ Change the water flow rates during a time period (t~ In). For time events to < t I < . . . ( tn, measure data about surface temperatures, superheat, casting speed, mold heat flux, and spray water flow rates. (2) Set time index k = 0. (3) Increase the time by setting k = k + 1. Set initial guess for the heat-transfer coefficient hi = h k. (4) Input the at time tk measured process data to the developed simulation model and calculate the surface temperature at time t~. (5) IF the calculated temperature does not match the measured temperature, T H E N change the heat-transfer coefficient hi slightly and goto (4), ELSE save hi and Wa at time tk. (6) If k < n goto (3) (new time step begin). (7) Apply a standard curve fitting method for solving the parameters aa and fla from Eq. [22] using the calculated time series of (hi(t l) . . . . . hi(in)) and measured time series of (Wa(t 1). . . . . Wa(in)) This simulation procedure is generally repeated separately for each zone where measurements are taken, starting from the zone nearest to the mold. This is because the parameters for the previous zones must always be known. The flow chart is shown in Figure 9.

Step 2

[22] Here a~, fli, and % are parameters for zone i. The parameter % is a function of temperature, and it takes into account the effect of the strand surface temperature on the heat-transfer coefficient. The strand surface temperature has little effect on h when it is higher than the Leidenfrost temperature and the parameter % = %(T) is set equal to one. The parameters oli, ~i, and 7, were derived in two steps. In the first step, o~i and fla were derived from measurements done under conditions in which 7i could be set equal to 1. In the second step, for temperatures below the Leidenfrost temperature, % was derived using values determined in step 1 for aa and /~i" These two steps are described subsequently.

] I ZONE-ZONE+I

I

sm,.0

IJ

p

I

I STF.P-STEP+I p1.1 I SET INmALGUESS FOR h ] I ~LKIE'I-r.MPI~kTUREDISTRIBUTION

[.,, p

Step 1 (1) Assume that the process is under steady state at time

1

"~ ' ~ ' = ~ m~-1-..........~ ~b~ ~"-ik.~ =o 0.6 ro

biondinear fs

~

I

no

s,v h.W

CHANGEh

[

I

(calculated)

Used approximation in the model

\

% \

-=o 04 0.2

~x

I CURVEe,I lING: h(W)=o~WI~

(TI = 1455, Ts = 1396, Tp = 1434) 0.0 1395

I

t

I

I

I

1405

1415

1425

1435

1445

1455

Temperature, *C Fig. 8 - - S o l i d fraction vs temperature during solidification for AISI 304L calculated by the solidification model. 139~ 6901VOLUME 24B, AUGUST 1993

Fig. 9 - - F l o w chart showing the determination of the parameters ai and /3~ in the correlation hi = a~W~,~ or h~ oliW i -~ ~i. =

METALLURGICAL TRANSACTIONS B

In this step, the parameter %(T) was determined experimentally by fitting the calculated temperatures to the measured ones. During these measurements, the spray water flow rates were kept constant but the casting speed was changed so that the strand surface temperature fell below the Leidenfrost temperature. The parameters ai and /3i were the same are were derived beforehand in step 1. Figure 10 shows the variation of y with temperature. In our test calculations, this was used for each zone. The parameters ai and/3i obtained in step 1 are presented in Section VIII.

VIII. COMPARISON OF CALCULATED RESULTS WITH MEASURED VALUES To test the accuracy of the model, surface temperatures measured with pyrometers were compared with the calculated values obtained by the model. The measurements were performed with a stainless steel slab caster. Some results between the surface temperature calculated by the model and measured by the pyrometers are shown in the Figures 11 through 13. The pyrometer data presented are 15-second mean values of the measured 1second temperatures. It is assumed that the fluctuations in surface temperature are caused by the scale on the surface of the slab, and the top temperatures are near the actual surface temperatures. In Figure 11, the temperatures in zone 3 at 2.7 m below the mold on the bottom side of the strand are compared. The steel grade was AISI 304L. The following correlation was used for the heat-transfer coefficient for the zones i = 1, 2, and 3: ( J hi

0.019 W~/74 ~ T )

1

((0.003

W i ~"

0.21) y(r)

Wi~ 6 0 - m 2 min l Wi> 6 0 - m 2 min

[23]

steel grade was AISI 321. The correlation for the heattransfer coefficient for the zones i = 1, 2, and 3 was now the following: l

l 0.0175 W~FTM T(T)

W, <- 60 m2 mi-----~

hi =

[24]

l Wi > 60 me min

L(0.003 Wi + 0.18) y(T)

The casting speed was kept constant during this trial except at the end of the casting, but the water flow rates were increased suddenly by 40 pet during the casting (Figure 12). Quite good agreement between the calculated and measured results is again obtained. In Figure 13, the comparison point is in zone 4, 5.8 m below the mold on the bottom side of the strand. The steel grade was AISI 316L. Equation [23] was used for the first three zones, but for the fourth zone, the following correlation was used: 0.021 W~474 ")/(Z)

W4 <-- 60

h4=

I m: min l

(0.003 W4 + 0.25) N T )

W4 > 60 m2 rain

[25] In this case, the strand surface temperature is quite low and there are high fluctuations in the measured temperature. It is difficult to know if the fluctuation is a measurement error or not. Error could be caused, for instance, due to scale on the surface of the slab. Our opinion is that the fluctuation is caused due to the error in measurements (scale), and the top temperatures are close to the actual surface temperatures. So, the agreement between the results is again quite good.

1200

The casting speed was changed during this trial, but the water flow rates were constant. As Figure 11 shows, quite a good agreement between the calculated and measured values is obtained. In Figure 12, the comparison point is the same as in Figure 11 but now on the top side of the strand. The

1100

oO 1 0 0 0

2 G) D.

90o

E

8oo

~o

700

3.0

,,,,z

Casting

r~ 60o ~"

speed

2.0 m/min)

50O

1.5 1.0

,

,

400

,

0 0.5

-600

J 650

700

750

800

850

q

;

900

950

Temperature, ~

Fig. 1 0 - - T h e parameter y(T) obtained from the measurements. METALLURGICAL TRANSACTIONS B

1000

[

I

I

I

I

I

I

I

10

20

3O

4O

5O

60

70

8O

9O

~me, min Fig. 11 - - C o m p a r i s o n between calculated and measured surface temperatures for steel grade AISI 304L with change in casting speed. The comparison point is 2.7 m below the mold on the bottom side. VOLUME 24B, AUGUST 1993--691

1200

oC1 1100

ql t,n

100o

E

Measured

90o o -,I

8O0 .~asting speed = 0.96 m/rain, except at the end 700 O

t

t

I

F

I

10

20

30

40

50

60

Time, rain Fig. 1 2 - - C o m p a r i s o n between calculated and measured surface temperatures for the steel grade AISI 321 with change in the spray water flow rates. The comparison point is 2.7 m below the mold on the top side. 1000 Calculated

9o0 coo r,

E

G) q) o

*"

700

Casting speed

500 0

m/min, except at the end

= 0.97

F

i

i

i

I

i

i

i

!

10

20

30

40

50

60

70

80

90

b 100 110

Time, min Fig. 1 3 - Comparison between calculated and measured surface temperatures for the steel grade AISI 316L. The comparison point is 5.8 m below the mold on the bottom side.

The presented equations for hi (Eqs. [23] through [25]) are derived from measurements with W = 15 - 50 1/ m 2 min. Outside this range, the equations were estimated from the literature. IX.

DISCUSSION

Good agreement between the calculated and measured surface temperatures was obtained, as Figures 11 through 13 show. However, in some cases, the calculated slab surface temperatures were higher than the measurements for approximately 10 to 30 minutes after the start of the cast, as seen in Figure 13. A probable reason for this is that the casting machine rolls and spray cooling water heat up to a steady state condition after start-up. This phenomenon was not taken into account in the model, but it is now being incorporated into the model. The shell thickness was not compared with experimental values. Some experimental shell thickness measurements have 692--VOLUME 24B, AUGUST 1993

recently been carried out, and the analysis of them is currently in progress. The heat-transfer coefficient depends also on the degree of the oxidation on the surface. [26,4~ In this study, three different steel grades (AISI 304L, AISI 321, AISI 316L) were used, and AISI 321 especially had a different dependence on the heat-transfer coefficient than the other two grades. AISI 321 is Ti-stabilized and was cast with different casting powder than the other two grades. The correlation between the cooling conditions and the heat-transfer coefficient must be determined separately for each steel group. There were also situations where the use of an average heat-transfer coefficient for the cooling zones was not accurate enough. For instance, when the casting speed is zero, the same areas of the slab are sprayed by the cooling water while the unsprayed areas may reheat. Such a period cannot be simulated very accurately using average heat-transfer coefficients for the cooling zones. In such cases, the heat-transfer coefficients should be determined within the cooling zones locally more precisely 17-1o] The aim of the present work was to develop a model which calculates the strand surface temperature profile and the shell thickness profile in the middle region of the strand on-line for process control purposes. In summary, it can be said that the model gives reliable results if the boundary conditions are correctly determined. It also fulfills the requirements for on-line use. As to the future work, the main aim is to study more precisely the dependence of the heat-transfer coefficient on the cooling conditions (water spray, air mist cooling, steel grade, etc.) using both pyrometer and thermoelement measurements. Another aim for future work is the development of on-line applications of the model as a control of the secondary cooling, on-line quality prediction, or optimization of the casting speed. The model is now implemented on a one slab casting machine. X.

CONCLUSIONS

A real-time heat-transfer model for continuous slab casting is presented. The model calculates the strand temperatures and the solid shell thickness profile along the machine as a function of the actual casting variables, strand geometry, and steel grade. The model was tested by carrying out industrial trials, and quite a good agreement between the calculated and measured results was obtained. The model also seems to fulfill the speed requirements concerning its on-line use. A special procedure to determine the boundary conditions for the secondary cooling zones from temperature measurements is also developed and presented. ACKNOWLEDGMENTS The authors wish to thank Professor Lauri Holappa at the Helsinki University of Technology and the Technology Development Center for financial support.

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